Vanna-Volga methods applied to FX derivatives: a practitioner s approach

Transcription

1 Vanna-Volga methods applied to FX derivatives: a practitioner s approach F. Bossens ARAB-KVBA Summer School 7-8 September

2 Presentation layout Problem statement: o Vanilla prices : the implied volatility surface o pricing exotic options : smile dynamics o Comparing prices from different models The Vanna-Volga Method o The general idea o Basic implementation for vanillas o Adaptations for the pricing of exotic options o A concrete calibration example Conclusions Bibliography ARAB-KVBA Summer School 7-8 September

3 Problem statement : the implied smile Price of Vanillas (Call / Put) uniquely determined by the Implied volatility surface : Market Implied Vol Surface Black-Scholes formula σ 12.0% 11.0% 10.0% vol 9.0% 8.0% 7.0% 6.0% spot strike %-12.0% 10.0%-11.0% 9.0%-10.0% 8.0%-9.0% 7.0%-8.0% 6.0%-7.0% 05/02/ /12/ /10/ /08/ /06/ /05/ /03/2008 time Expiry c d = 1 d 2 S = 0 = e qt ln( S 0 ln( S N ( d ) / K ) + ( r q + σ ( K, T ) / K ) + K e rt σ ( K, T ) ( r q σ ( K, T ) 2 σ ( K, T ) N ( d Terminal spot density: 2 rt d f ( S, T ) = e Call ( S, T, K, σ ( K )) 2 dk T T 2 ) / 2) T / 2) T K = S High-amplitude movements more likely than predicted by a normal distribution ARAB-KVBA Summer School 7-8 September

4 Problem statement : smile dynamics Exotic options (e.g. calls/puts with barriers) : payout depends on entire spot path S(t), 0<t<T Price not only function terminal density (vol surface), but also on transition density (=smile dynamics) f(s 1,T 1,S 2,T 2 ) = density of spot at time T 2, conditional on spot starting from S 1 at time T 1. Market Implied Vol Surface 11.0%-12.0% 10.0%-11.0% 9.0%-10.0% 12.0% 11.0% 10.0% vol 9.0% 8.0% 7.0% 6.0% 8.0%-9.0% 7.0%-8.0% 6.0%-7.0% 05/02/ /12/ /10/ /08/ /06/2008 time Problem : transition densities are not fully determined even by the complete knowledge of the entire implied volatility surface spot /05/ /03/2008 ARAB-KVBA Summer School 7-8 September

5 Problem statement : smile dynamics Various stochastic models can be perfectly calibrated to the same implied smile (they all give the exact same price for vanilla options) and yet yield very different prices for exotic options. Schoutens, W., Simons, E. and Tistaert, J., A Perfect calibration! Now what?, Wilmott Magazine, March 2004 ARAB-KVBA Summer School 7-8 September

6 Problem statement : smile dynamics (cont d) Example : 2 very different smile dynamics d S Pure local volatility (Dupire) Vol = μ S dt +σ( S, t) S dw Implied volatility Smile Maturity (USDCHF 8 Jan.09) Maturity : 1y Market market vol Loc. Heston Vol vol (analytic) Heston Vol (MC) Stoch Vol ds dv E Pure stoch. volatility (Heston) = μs dt = κ( V [ dw dw ] V S V ) dt = ρdt dw + ε 1 V dw The 2 models are able to fit vanilla prices (after calibration of Heston) But Strike ARAB-KVBA Summer School 7-8 September

7 Which model is better? Test on the pricing of a One-Touch option Actual smile dynamics seems to follow a mixture between those 2 extreme regimes ARAB-KVBA Summer School 7-8 September

8 A tradeoff A rigorous mixture model Stochastic vol / local vol / jumps is certainly able to produce correct market prices, however Complexity of implementation Delicate calibration Computationally demanding Alternatively, the «Vanna-Volga» follows a more «cooking recipe» approach Strictly speaking, Vanna-Volga is not a model, rather a price adjustment method Easy implementation Simple / No calibration Computationally very efficient ARAB-KVBA Summer School 7-8 September

9 Vanna-Volga : main idea Derives from the trader s idea that the smile adjustment to an option price is associated with costs incurred by hedging its volatility risk Build a portfolio of 3 vanilla strategies, which zeros out the vega ( P/ σ), Volga ( ²P/ σ²) and Vanna ( ²P/ σ S) of the option at hand. Using the market-implied volatility surface, calculate the smile impact on the obtained portfolio of vanillas. Market Price (Exotic Option) = Black-Scholes Price (Exotic Option) + Smile Impact (vanilla hedging portfolio) ARAB-KVBA Summer School 7-8 September

10 Vanna-Volga : the vanilla hedging portfolio Composed of 3 liquid vanilla strategies: At-The-Money (ATM) = ½ Call(K ATM S 0 ) + ½ Put (K ATM S 0 ) 25Δ Risk-Reversal (RR) = Call(Δ = 0.25) - Put(Δ = -0.25) 25Δ Butterfly (BF) = ½ Call(Δ = 0.25) + ½ Put(Δ = -0.25) - ATM Profit ATM = ½ Straddle RR BF K ATM S 0 S T K 25P K ATM K 25P K ATM K 25C K 25C S T S T ARAB-KVBA Summer School 7-8 September

11 Vanna-Volga : the vanilla hedging portfolio selecting the weights: Price of exotic option : X Price of hedging portfolio : P = w ATM ATM + w RR RR + w BF BF X and P computed under Black-Scholes assumptions (no smile) Zeroing out the sensitivity to volatility, up to 2 nd order (Vega, Vanna and Volga) : w r A 1r = x ARAB-KVBA Summer School 7-8 September

12 Vanna-Volga: full correction Finally the «Vanna volga» price of an option is : X VV = X BS +w ATM (ATM Mkt -ATM BS ) + w RR (RR Mkt -RR BS ) + w BF (BF Mkt -BF BS ) Black-Scholes Price Smile Impact on vanilla hedging portfolio Implied Volatility Mkt BS K ATM ARAB-KVBA Summer School 7-8 September

13 Vanna-Volga: full correction Applying the full VV correction to price a vanilla provides a reasonable approximation of the original smile : Applying the full VV correction to price an exotic option usually over-estimates the smile impact : ARAB-KVBA Summer School 7-8 September

14 Vanna-Volga: attenuating the correction Starting from the original VV setting : X VV = X BS + w RR (RR Mkt -RR BS ) + w BF (BF Mkt -BF BS ) Apply some simple matrix calculus, drop vega contribution and introduce attenuation factors: X VV = X BS + p VANNA (γ) X VANNA Ω VANNA + p VOLGA (γ) X VOLGA Ω VOLGA Market practice : p VANNA and p VOLGA are functions of o γ = Survival probability T} o γ = Expected first exit time (FET) / obtained by solving BS PDE with appropriate boundary condition ARAB-KVBA Summer School 7-8 September

15 Vanna-Volga: requirements on attenuation factors 0 < p VANNA (γ), p VANNA (γ) < 1 As spot level approaches a Knock-out barrier, Vanna-Volga correction should fade-out o For volga term, always true as volga fades-out: o Not the case for Vanna, hence, we should impose: lim γ 0 p VANNA ( γ ) = 0 lim X VOLGA γ 0 = 0 Vanilla payouts (no barriers) are best priced with the full Vanna-Volga correction: lim p VANNA ( γ ) = lim p VOLGA ( γ ) = 1 γ 1 γ 1 ARAB-KVBA Summer School 7-8 September

16 Vanna-Volga: example of attenuation factors choice p VANNA (γ) p VOLGA (γ) Composite linear function a γ (Survival probability or FET) a? Calibration from market prices of selected exotic options (typically touch options) ARAB-KVBA Summer School 7-8 September

17 Vanna-Volga: example of calibration One-Touch Call with barrier a=0.42 Call with 2 barriers Calibration on OT yields satisfactorily results for other exotic instruments Optimized Vanna-Volga approach yields better prices than Heston or Local-Vol ARAB-KVBA Summer School 7-8 September

18 Concluding remarks Price of Exotic (Path-dependant) options: o not uniquely determined by knowledge of vanilla prices o depends on smile dynamics (model) In FX markets, Vanna-Volga is an appealing alternative to more rigorous stochastic models o Easy implementation o Simple calibration o Computationally very efficient Limitations o Not a model, impossible to simulate dynamics Limited to options for which the survival probability or FET can be computed o Correction based on smile at maturity date only, not on the entire volatility surface o Does not account for stochastic interest rates effects Maturity <1 2Years o Does not account for jumps in the underlying process Maturity >1 Month ARAB-KVBA Summer School 7-8 September

19 Bibliography F. Bossens, G. Rayée, N. S. Skantzos and G. Deelstra, Vanna-Volga methods applied to FX derivatives : from theory to market practice Available at (submitted to International Journal of Theoretical and Applied Finance) A. Castagna and F. Mercurio, The Vanna-Volga method for implied volatilities, Risk (January 2007) W. Schoutens, E. Simons, and J. Tistaert, A Perfect calibration! Now what?, Wilmott Magazine (March 2004) H. J. Stein, FX Market Behavior and Valuation, Available at SSRN: (December 13, 2006) U. Wystup, Vanna-Volga Pricing, MathFinance AG (June 2008) Available at ARAB-KVBA Summer School 7-8 September